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Perturbation methods in group analysis. (English. Russian original) Zbl 0759.35003
J. Sov. Math. 55, No. 1, 1450-1490 (1991); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Noveishie Dostizh. 34, 85-147 (1989).
See the review in Zbl 0722.35005.

MSC:
35A22 Transform methods (e.g., integral transforms) applied to PDEs
35A25 Other special methods applied to PDEs
58J72 Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
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References:
[1] M. Ablowitz and H. Sigur, Solitons and the Method of the Inverse Problem [Russian translation], Mir, Moscow (1987).
[2] V. A. Baikov, R. K. Gazizov, and N. Kh. Ibragimov, ?Approximate symmetries of equations with small parameter?, Preprint Inst. Prikl. Mat., AN SSSR, Moscow (1987), No. 150.
[3] V. A. Baikov, R. K. Gazizov, and N. Kh. Ibragimov, ?Formal symmetries and Bäcklund transformations?, Preprint Inst. Prikl. Mat. AN SSSR, Moscow (1987), No. 226.
[4] V. A. Baikov, R. K. Gazizov, and N. Kh. Ibragimov, ?Approximate group analysis of the equation utt?(f(u)ux)x+??(u)ut=0?, Preprint, Inst. Prikl. Mat. AN SSSR, Moscow (1988), No. 68.
[5] V. A. Baikov, R. K. Gazizov, and N. Kh. Ibragimov, ?Approximate symmetries?, Mat. Sb.,136, No. 4, 435?450 (1988).
[6] V. A. Baikov, R. K. Gazizov, and N. Kh. Ibragimov, ?Approximate group analysis of the nonlinear equation utt?(f(u)ux)x+??(u)ut=0?, Differents. Uravn., No. 7, 1127?1138 (1988).
[7] V. A. Baikov, R. K. Gazizov, and N. Kh. Ibragimov, ?Linearization and formal symmetries of the Korteweg-de Vries equation?, Dokl. Akad. Nauk SSSR,303, No. 4, 781?784 (1989).
[8] N. Kh. Ibragimov, Transformation Groups in Mathematical Physics [in Russian], Nauka, Moscow (1983).
[9] N. Kh. Ibragimov and A. B. Shabat, ?Korteweg-de Vries equation from a group point of view?, Dokl. Akad. Nauk SSSR,244, No. 1, 57?61 (1979). · Zbl 0423.35076
[10] V. L. Katkov, ?Group classification of solutions of Hopf’s equation?, Zh. Prikl. Mekh. Tekh. Fiz., No. 6, 105?106 (1965).
[11] R. Courant, Partial Differential Equations [Russian translation], Mir, Moscow (1964).
[12] J. L. Lem, Introduction to the Theory of Solitons [Russian translation], Mir, Moscow (1983).
[13] A. H. Naife, Methods of Perturbations [Russian translation], Mir, Moscow (1976).
[14] L. V. Ovsyannikov, Group Properties of Differential Equations [in Russian], Izd. Sib. Otdel. Akad. Nauk SSSR, Novosibirsk (1962).
[15] L. V. Ovsyannikov, ?Group properties of equations of mechanics?, in: Mechanics of Continuous Media and Related Problems of Analysis [in Russian], Nauka, Moscow (1972).
[16] L. V. Ovsyannikov, Group Analysis of Differential Equations [in Russian], Nauka, Moscow (1978). · Zbl 0484.58001
[17] A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov, Regimes with Straightening in Problems for Quasilinear Parabolic Equations [in Russian], Nauka, Moscow (1987).
[18] A. A. Samarskii, N. V. Zmitrienko, S. P. Kurdyumov, and A. P. Mikhailov, ?Effect of metastable localization of heat in a medium with nonliner thermal conductivity?, Dokl. Akad. Nauk SSSR,223, No. 6, 1344?1347 (1975).
[19] A. N. Tikhonov, A. B. Vasil’eva, and A. G. Sveshnikov, Differential Equations [in Russian], Nauka, Moscow (1980).
[20] I. A. Charnyi, Unstable Motion of a Real Fluid in Pipes [in Russian], Nedra, Moscow (1975).
[21] B. V. Shabat, Introduction to Complex Analysis. Part 2. Functions of Several Variables [in Russian], Nedra, Moscow (1985). · Zbl 0578.32001
[22] W. F. Ames, E. Adams, and R. J. Lohner, ?Group properties of utt=[f(u)ux]x?, Int. J. Non-Linear Mech.,16, Nos. 5?6, 439?447. · Zbl 0503.35058
[23] M. Flato, G. Pinczon, and J. Simon, ?Non-linear representation of Lie groups?, Ann. Scient. Ecole Norm. Sup., Paris,10, No. 4, 405?418 (1977). · Zbl 0384.22005 · doi:10.24033/asens.1331
[24] R. Hirota, ?Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons?, Phys. Rev. Lett.,27, 1192?1194 (1971). · Zbl 1168.35423 · doi:10.1103/PhysRevLett.27.1192
[25] R. Hirota, ?Exact solution of the modified Korteweg-de Vries equation for multiple collisions of solitons?, J. Phys. Soc. Jpn.,33, No. 3, 1455?1458 (1972).
[26] R. Hirota, ?Nonlinear partial difference equation. I. A difference analogue of the Korteweg-de Vreis equation?, J. Phys. Soc. Jpn.,43, No. 4, 1424?1433 (1977). · Zbl 1334.39013 · doi:10.1143/JPSJ.43.1424
[27] N. H. Ibragimov, ?Sur l’equivalence des equations d’evolution, qui admettent une algebre de Lie?Bäcklund infinie?, C. R. Acad, Sci., Ser. I, Paris,293, No. 14, 657?660 (1981).
[28] S. Lie and F. Engel, Theorie der Transformationsgruppen, Teubner, Leipzig, Vol. 1 (1888), Vol. 2 (1890), Vol. 3 (1893).
[29] R. R. Rosales, ?Exact solutions of some nonlinear evolution equations?, Stud. Appl. Math.,59, 117?151 (1978). · Zbl 0387.35061 · doi:10.1002/sapm1978592117
[30] E. Taflin, ?Analytic linearisation of the Korteweg-de Vries equation?, Pac. J. Math., 108, No. 1, 203?220 (1983). · Zbl 0469.35078 · doi:10.2140/pjm.1983.108.203
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